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TECHNICAL PAPERS: Soft Tissue

A Lagrange Multiplier Mixed Finite Element Formulation for Three-Dimensional Contact of Biphasic Tissues

[+] Author and Article Information
Taiseung Yang

Department of Mechanical, Aeronautical and Nuclear Engineering, and Scientific Computation Research Center, Rensselaer Polytechnic Institute, Troy, NY

Robert L. Spilker1

Department of Mechanical, Aeronautical and Nuclear Engineering, Department of Biomedical Engineering, and Scientific Computation Research Center, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590spilker@rpi.edu

1

Corresponding author.

J Biomech Eng 129(3), 457-471 (Oct 25, 2006) (15 pages) doi:10.1115/1.2737056 History: Received April 14, 2006; Revised October 25, 2006

A three-dimensional (3D) contact finite element formulation has been developed for biological soft tissue-to-tissue contact analysis. The linear biphasic theory of Mow, Holmes, and Lai (1984, J. Biomech., 17(5), pp. 377–394) based on continuum mixture theory, is adopted to describe the hydrated soft tissue as a continuum of solid and fluid phases. Four contact continuity conditions derived for biphasic mixtures by Hou (1989, ASME J. Biomech. Eng., 111(1), pp. 78–87) are introduced on the assumed contact surface, and a weighted residual method has been used to derive a mixed velocity-pressure finite element contact formulation. The Lagrange multiplier method is used to enforce two of the four contact continuity conditions, while the other two conditions are introduced directly into the weighted residual statement. Alternate formulations are possible, which differ in the choice of continuity conditions that are enforced with Lagrange multipliers. Primary attention is focused on a formulation that enforces the normal solid traction and relative fluid flow continuity conditions on the contact surface using Lagrange multipliers. An alternate approach, in which the multipliers enforce normal solid traction and pressure continuity conditions, is also discussed. The contact nonlinearity is treated with an iterative algorithm, where the assumed area is either extended or reduced based on the validity of the solution relative to contact conditions. The resulting first-order system of equations is solved in time using the generalized finite difference scheme. The formulation is validated by a series of increasingly complex canonical problems, including the confined and unconfined compression, the Hertz contact problem, and two biphasic indentation tests. As a clinical demonstration of the capability of the contact analysis, the gleno-humeral joint contact of human shoulders is analyzed using an idealized 3D geometry. In the joint, both glenoid and humeral head cartilage experience maximum tensile and compressive stresses are at the cartilage-bone interface, away from the center of the contact area.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Finite element discretization with tetrahedral biphasic elements and triangular contact elements for the 3D v-p contact formulation (solid circles represent solid velocity DOFs, and hollow circles represent pressure DOFs). Interpolation for the Lagrange multipliers, λf and λs, are discontinuous constant and linear functions, respectively

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Figure 2

Closest-point projection from a Gauss quadrature point on the contactor surface (on body A) to the target surface (on body B)

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Figure 3

(a) Schematic of the biphasic indentation test with a flat-ended cylindrical indenter and (b) the finite element mesh with 14647 tetrahedral elements and 294 contact elements

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Figure 4

(a) Axial strain, (b) axial stress, and (c) pressure along the radial position at several depths at t=500s in the biphasic indentation test with a flat-ended cylindrical indenter

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Figure 5

Evaluation of the distribution of pressure (a), axial normal stress (b), and relative axial fluid velocity (c) at t=250s in the biphasic indentation test with a flat-ended cylindrical indenter. The 3D distribution in the tissue and indenter is shown on the left, and the continuity between the two bodies along the contact surface is shown on the right

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Figure 6

Schematic of the biphasic indentation test with a cylindrical ended indenter

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Figure 7

Axial displacement and pressure distributions on the deformed configuration for the biphasic indentation at t=100s in the biphasic indentation test with a cylindrical ended indenter

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Figure 8

Normal traction distributions (in mega Pascal) at t=100s in the biphasic indentation test with a cylindrical ended indenter

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Figure 9

Model geometry and a finite element mesh for glenoid and humeral head cartilage layers

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Figure 10

Undeformed geometry in the GHJ, and the displacement on the deformed geometry at t=5s (0.1mm applied compressive displacement) and t=10s (0.2mm applied compressive displacement)

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Figure 11

3D pressure distribution (in mega Pascal) in the GHJ on the deformed geometry (left) and at the tissue-bone interfaces (right) at t=10s (0.2mm applied compressive displacement)

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Figure 12

Distribution of maximum (a) and minimum (b) principal elastic stress (in mega Pascal) in the GHJ (left) and the value at the tissue-bone interfaces (right) at t=10s (0.2mm applied compressive displacement)

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